Calculus

Applications of the Derivative

Evolute and Involute

Let a plane curve \(\gamma\) be given by the natural equation

\[\mathbf{r} = \mathbf{r}\left( s \right),\]

where the parameter \(s\) means the arc length of the curve. Suppose that at each point, the curvature of the curve is not zero: \(K\left( s \right) \ne 0.\) Then at any point \(M\) we can define a finite radius of curvature:

\[R = R\left( s \right) = \frac{1}{{K\left( s \right)}}.\]

On the normal \(\mathbf{n}\) we draw the segment \(\mathbf{MC}\) equal to the radius of curvature \(R\left( s \right)\) at the point \(M\) (Figure \(1\)).

Figure 1.

The point \(C\) is called the center of curvature of the curve \(\gamma\) at point \(M.\)

If the radius vector of the center of curvature is denoted by \(\require{AMSmath.js}\boldsymbol{\rho},\) then

Note that the condition of non-zero curvature at all points of the curve is rigid enough. As a result, certain curves, for example, with inflection points are excluded from analysis. Therefore, sometimes a more general case of arbitrary curvature is considered. If the curvature at a point is zero, the evolute at this point has a discontinuity. Such case is shown schematically in Figure \(2.\)